The existence and stability of discrete solitons in waveguide arrays exhibiting a linear variation of the effective index and a Kerr nonlinearity is studied. We find that the resonant coupling of the conventional discrete soliton to a linear Wannier-Stark state does not entail soliton decay. We rather observe the formation of a bound state where the Wannier-Stark state gets nonlinearly modified. This results in an infinite number of isolated branches of hybrid discrete solitons.